Source-linked AI summary
Fully Distributed Event-Triggered Protocols for Linear Multi-Agent Networks
Bin Cheng, Zhongkui Li
TL;DR
The paper tackles event-triggered consensus for general linear multi-agent networks without relying on continuous communication or global graph information. It designs adaptive protocols from sampled local state or output information for leaderless and leader-follower settings, and reports consensus with Zeno behavior excluded. The protocols are fully distributed and scalable, with design independent of network scale.
Problem
Event-triggered consensus requires distributed control laws and triggering functions that achieve agreement while avoiding Zeno behavior under constrained communication.
Method
The paper designs distributed adaptive event-triggered protocols using sampled local state information or local output information for leaderless and leader-follower consensus.
Results
The proposed protocols ensure consensus, exclude Zeno behavior, and do not require continuous communications for control-law updating or triggering-function monitoring.
Takeaways & Limitations
The protocols are fully distributed and scalable because they use no global network-graph information and are independent of network scale.
Abstract
from arXiv · showhide
This paper considers the distributed event-triggered consensus problem for general linear multi-agent networks. Both the leaderless and leader-follower consensus problems are considered. Based on the local sampled state or local output information, distributed adaptive event-triggered protocols are designed, which can ensure that consensus of the agents is achieved and the Zeno behavior is excluded by showing that the interval between any two triggering events is lower bounded by a strictly positive value. Compared to the previous related works, our main contribution is that the proposed adaptive event-based protocols are fully distributed and scalable, which do not rely on any global information of the network graph and are independent of the network's scale. In these event-based protocols, continuous communications are not required for either control laws updating or triggering functions monitoring.
I. INTRODUCTION
The paper addresses event-triggered consensus for general linear multi-agent networks under communication and power constraints. It develops adaptive protocols intended to be fully distributed, scalable, and applicable to both leaderless and leader-follower settings.
- Motivation: Communication bandwidth and agent power constraints motivate reducing the continuous information exchange required by traditional networked control strategies.Cooperative control depends on information exchange among neighboring agents, while many practical systems are wireless and spatially distributed.
- Related work: Existing event-triggered consensus studies include distributed, decentralized, and observer-based protocols for several agent models and network settings.The cited studies address single- and double-integrator systems, linear multi-agent networks, and leader-follower consensus.
- Research gap: Prior protocols generally require global network information, such as nonzero Laplacian eigenvalues, to select parameters in control laws or triggering functions.The paper notes that estimating such information from network size is incompatible with scalable design.
- Contributions: The proposed protocols target leaderless and leader-follower consensus for general linear networks using sampled local state or output information.The leaderless state-based protocol includes time-varying weights in both control laws and triggering functions, while an observer-based protocol uses local output information.
- Contributions: The protocols are fully distributed and scalable, requiring neither global graph information nor dependence on the network’s scale.The paper also states that continuous communications are not required for control-law updating or triggering-function monitoring.
II. PROBLEM STATEMENT AND MOTIVATIONS
The paper formulates event-triggered consensus for identical continuous-time general linear agents communicating over a graph. The objective is asymptotic agreement while excluding infinitely many events in finite time through locally sampled control and triggering mechanisms.
- Agent model: Each agent is modeled by identical continuous-time general linear dynamics with state x_i, control input u_i, and output y_i.The state, input, and output dimensions are n, p, and q, respectively.
- Communication graph: The communication network is represented by a directed graph G=(V,E), with adjacency and Laplacian matrices describing neighbor relationships and graph structure.The passages define directed paths, spanning trees, undirected graphs, connectivity, adjacency entries, and Laplacian entries.
- Communication graph: For an undirected graph, zero is a simple Laplacian eigenvalue exactly when the graph is connected.The smallest nonzero Laplacian eigenvalue is characterized through a constrained Rayleigh quotient.
- Objective: The consensus objective is lim t→∞ ||x_i − x_j|| = 0 for all agents, while Zeno exclusion requires no infinite number of events within finite time.Control laws use local information sampled at discrete event times, and triggering functions determine when agents broadcast their states.
III. FULLY DISTRIBUTED EVENT-BASED PROTOCOLS FOR LEADERLESS CONSENSUS
For leaderless consensus, the paper designs fully distributed event-triggered protocols for connected undirected graphs under stabilizability, detectability, and graph-connectivity assumptions.
- Protocol design: The section presents fully distributed event-triggered protocols for leaderless graphs.The design is introduced after stating the required assumption.
- Assumptions: The leaderless protocol design assumes an undirected connected communication graph and a stabilizable and detectable agent model.These conditions are stated in Assumption 1.
A. State-Based Adaptive Event-Triggered Protocols
The paper develops a state-based adaptive event-triggered protocol using sampled neighbor states, time-varying coupling weights, and a hybrid triggering function. Under stated conditions, the protocol bounds or eliminates consensus error, excludes Zeno behavior, and requires no global graph information.
- Protocol design: The protocol uses sampled local states and defines measurement error as the difference between each agent’s sampled and current state.Controllers update at event instants and when neighbors broadcast new states.
- Protocol design: Time-varying coupling weights enter both the control law and triggering function, producing a hybrid trigger with state-dependent and exponentially decaying terms.The design relies on sampled states rather than continuous neighbor-state monitoring.
- Protocol design: Algorithm 1 computes P from an algebraic Riccati equation, sets K = −B^T P and Γ = PBB^T P, and selects positive protocol parameters.Theorem 1 requires (A, B) to be stabilizable for the stated Riccati-equation construction.
- Consensus guarantees: Theorem 1 establishes uniform ultimate boundedness of the consensus error and coupling gains, with exponential convergence to an arbitrarily small residual set under suitable parameter conditions.With the sigma-modification term removed, Corollary 1 gives asymptotic convergence of the consensus error to zero.
- Consensus guarantees: The protocol is robust to bounded external disturbances, although the resulting consensus-error bound depends on the disturbance bounds and protocol parameters.The disturbance robustness argument is described as following similar proof steps, with details omitted.
- Communication and triggering: Theorem 2 guarantees strictly positive inter-event intervals and excludes Zeno behavior, while the protocol remains fully distributed and scalable for connected topologies.The inter-event lower bound is generally conservative, depends on specific event times, and approaches zero as time tends to infinity.
B. Observer-Based Adaptive Event-Triggered Protocols
The observer-based protocol uses local output information with state estimates, adaptive event-triggered control, and triggering functions. Under the stated assumptions, consensus errors and coupling gains remain uniformly ultimately bounded, while Zeno behavior is excluded.
- The observer-based protocol is designed for the case where only local output information is available.
- Each agent uses an estimated state, sampled estimate, measurement error, and feedback gain matrices within the observer-based event-triggered protocol.The measurement error is defined from the sampled and current estimates.
- The observer gain F is chosen so that A+FC is Hurwitz, while the remaining design steps follow the state-based adaptive protocol.One admissible choice uses an algebraic Riccati equation.
- The consensus error and coupling gains are uniformly ultimately bounded, and the Zeno behavior is excluded under the observer-based adaptive protocol.The consensus-error conclusion follows from the observer error and auxiliary-state convergence arguments.
- The observer error ε asymptotically converges to zero because its Lyapunov derivative is negative definite.The derivative is given as −ωεTε for positive ω.
IV. FULLY DISTRIBUTED EVENT-BASED PROTOCOLS FOR LEADER-FOLLOWER CONSENSUS
For leader-follower consensus, followers use adaptive event-based control based on relative state information over a graph rooted at the leader. Under the stated assumptions, consensus errors and coupling gains are uniformly ultimately bounded and Zeno behavior is excluded.
- The leader-follower setting assumes a stabilizable pair (A, B), an undirected follower subgraph, and a directed spanning tree rooted at the leader.The leader is agent v1 and its control input is assumed to be zero.
- For each follower, the paper proposes an adaptive event-based control law and a corresponding triggering function using sampled relative-state information.The coupling weights vary over time and the measurement error is defined as ei = ˜xi − xi.
- Follower consensus errors are defined relative to the leader, whose state evolves according to the autonomous dynamics ˙x1 = Ax1.The leader's sampled state equals its actual state in this setting.
- Under Assumption 2, the consensus error and coupling gains are uniformly ultimately bounded, and the closed-loop system does not exhibit Zeno behavior.
V. SIMULATION EXAMPLES
The simulation section illustrates the theoretical results on linear multi-agent systems with randomly selected initial values and an undirected communication graph. The depicted graph satisfies Assumption 1.
- The simulations use linear multi-agent systems with the specified matrix A and randomly chosen initial values for all agents.
- The depicted communication graph satisfies Assumption 1.
- The communication topology is depicted as an undirected graph among the agents.
VI. CONCLUSION
The paper designs distributed adaptive event-based protocols for leaderless and leader-follower consensus in general linear multi-agent networks. These protocols are fully distributed and scalable, avoid continuous communications and Zeno behavior, and are illustrated through communication graphs, consensus errors, adaptive weights, and triggering instants.
- The paper designs distributed adaptive event-based protocols for both leaderless and leader-follower consensus in general linear multi-agent networks.
- The reported evaluation includes undirected communication graphs, consensus errors, adaptive coupling weights cij(t), and agents’ triggering instants.
- The proposed consensus protocols require no global network-graph information and are independent of the network’s scale.
- The event-triggered protocols do not require continuous communications among neighboring agents.
- The protocols exclude Zeno behavior.